Let a ,b ,c be such that b + c 0 if | * 20 c a& a + 1 & a - 1 - b… - Vidyadip

MCQ

Let $a ,b ,c $ be such that $b + c \ne 0$ if

$\left| {\begin{array}{*{20}{c}}a&{a + 1}&{a - 1}\\{ - b}&{b + 1}&{b - 1}\\c&{c - 1}&{c + 1}\end{array}} \right| + \left| {\begin{array}{*{20}{c}}{a + 1}&{b + 1}&{c - 1}\\{a - 1}&{b - 1}&{c + 1}\\{{{\left( { - 1} \right)}^{n + 2}} \cdot a}&{{{\left( { - 1} \right)}^{n + 1}} \cdot b}&{{{\left( { - 1} \right)}^n} \cdot c}\end{array}} \right| = 0$ then $n$ equals to

A
Zero
B
any even integer
✓
any odd integer
D
any integer

✓

Answer

Correct option: C.

any odd integer

c
$\left| {\begin{array}{*{20}{c}}
a&{a + 1}&{a - 1}\\
{ - b}&{b + 1}&{b - 1}\\
c&{c - 1}&{c + 1}
\end{array}} \right| + \left| {\begin{array}{*{20}{c}}
{{{\left( { - 1} \right)}^{n + 2}}a}&{a + 1}&{a - 1}\\
{{{\left( { - 1} \right)}^{n + 1}}b}&{b + 1}&{b - 1}\\
{{{\left( { - 1} \right)}^n}c}&{c - 1}&{c + 1}
\end{array}} \right|$

$ = \left| {\begin{array}{*{20}{c}}
{a + {{\left( { - 1} \right)}^{n + 2}}a}&{a + 1}&{a - 1}\\
{ - b + {{\left( { - 1} \right)}^{n + 1}}b}&{b + 1}&{b - 1}\\
{c + {{\left( { - 1} \right)}^n}c}&{c - 1}&{c + 1}
\end{array}} \right|$

$=0$ if $n$ is an obb integer.

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